Could you make the question more specific? At the moment you are basically asking for a list of things, which is bad in general. It would be much better if you'd split the question into several separate ones. Especially because some of the ways to measure certain aspects of the body have already received good answers here and don't need to be asked again.
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BaarnNov 25 '12 at 14:25

I am interested in measurements, but my questions is more complementary in that i am looking for assessments, not just single measurements.
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OlavNov 25 '12 at 16:14

Hi! I gave an answer, but didn't address "core", because I don't think it's an area of fitness. It contributes to the others though. Improved core strength will increase your balance, agility, ability to display power... maybe more!
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KateNov 25 '12 at 16:38

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The single measurement you're looking for is your VO2Max, but that's a very expensive test and needs specialized equipment. Just run the beep test. It is very correlated with VO2Max.
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KateNov 25 '12 at 17:23

1 Answer
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Here are some things you can do (I'm just making this up on the spot, based on somebody else's breakdown of the 10 areas of fitness, but they're things I've done in the past or currently to track my progress). They are mostly very standard ways to test these things and understood by any professional you'll visit.

Cardiovascular/respiratory endurance

Do the beep test. It's a good proxy for your VO2Max. All you need is to download the mp3.

How much can you power clean? Or, what is your vertical jump from a standstill? You can test this one by marking on a wall the highest spot you can touch while both feet are still planted on the ground. Then jump up as high as you can and mark the highest spot you can reach. The difference between the two is your "standing vertical jump" and is basically only improved by improving your power.

Speed

What is your 40 yard dash time?

Coordination

I can't think of a test for this one. Maybe something like bouncing a tennis ball off a wall and back to yourself (barehanded). Do this at a difficult distance. See how many times you can do this without dropping it.

Agility

The NFL combine's 3-cone drill and shuttle run are good agility tests:

The 3 cone drill tests an athlete's ability to change directions at a
high speed. Three cones in an L-shape. He starts from the starting
line, goes 5 yards to the first cone and back. Then, he turns, runs
around the second cone, runs a weave around the third cone, which is
the high point of the L, changes directions, comes back around that
second cone and finishes.

The short shuttle is the first of the cone drills. It is known as the
5-10-5. What it tests is the athlete's lateral quickness and explosion
in short areas. The athlete starts in the three-point stance, explodse
out 5 yards to his right, touches the line, goes back 10 yards to his
left, left hand touches the line, pivot, and he turns 5 more yards and
finishes.

Balance

There are a couple of ways to approach this. Do you want to test balance that you've acquired as a skill? Like standing one-legged on a balance board? Or do you want to see how your general training has helped you with balance in general? This is trickier... you'd have to see things like "am I rolling my ankles less while hiking", or "am I injured less during sports". But, since you want something repeatable, you could do a test like how long can you stand on a balance board, how long can you do it with one leg, with eyes closed, while moving your other leg to other points in space around the board, etc. But, don't practice whatever balance test you'll be doing other than in the tests, otherwise you'll just be testing a skill rather than your general balance ability.

Accuracy

This depends on what type of accuracy you want to test. Are you a thrower? Are you a volleyball setter? Are you a basketball shooter? Are you a hunter that shoots? You can pick whatever type of accuracy you would like to test and basically do what you'd do in the range with a gun to test your accuracy.

Throw a ball at a target on a wall. How close did you get? Repeat this 10 times and take the total. Or,

How many baskets do you make from 10 points around the key? Or,

What is your score at the shooting range? (This should be improved by improving your fitness!)

What you'll have

You might do this over a weekend once every 4 months or so, and get a results that look like this:

Accuracy: 95 (I don't know... some accuracy score, depending on what you do)

Then, you should make each of these numbers better for the next time you test, or you can bring them to a fitness professional and tell them which ones you want to improve in the next few months, and they can give you a program to suit that goal.